calculus of extensive quantities

8/6/2019 Calculus of extensive quantities

1/21

arXiv:1105

.3405v1[math.CT]17May2011

Calculus of extensive quantities

Anders Kock

University of Aarhus

Abstract. We show how a commutative monad gives rise to a theory of extensive quantities, includ-

ing (under suitable further conditions) a differential calculus of such. The relationship to Schwartz

distributions is dicussed. The paper is a companion to the authors Monads and extensive quantities,

but is phrased in more elementary terms.

IntroductionQuantities of a given type distributed over a given space (say distributions of smoke in a

given room) may often be added, and multiplied by real scalars ideally, they form a real

vector space. Lawvere stressed that the dependence of such vector spaces on the space

over which the quantities in question are distributed, should be taken into account; in fact,

the dependence is functorial. The viewpoint leads to a distinction between two kinds of

quantities: the functorality may be covariant, or it may be contravariant: In this context, the

covariant quantity types are called extensive quantities, and the contravariant ones intensive

quantities. This usage is an attempt to put mathematical precision into the use of these terms

in classical philosophy of physics. Mass distribution is an extensive quantity; mass density

is an intensive one. Lawvere observed that extensive and intensive quantites often come

in pairs, with a definite pattern of mutual relationship, like the homology and cohomology

functors on the category of topological spaces.In [14], we showed how such a pattern essentially comes about, whenever one has a

commutative monad T on a Cartesian Closed Category E (where E is meant to model some

category ofspaces, not specified further).

Such a monad is in particular a covariant endo-functor on E, so the emphasis in our

theory is the covariant aspect: the extensive quantities. We attempt to push a theory of these

as far as possible, with the intensive quantities in a secondary role.

This in particular applies to the differential calculus of extensive quantities on the line

R, which will be discussed in the last Sections 8 and 9; here, we also discuss the relationship

to the theory of Schwartz distributions of compact support (these have the covariant func-

torality requested for extensive quantities, and is a basis for a classical version of a theory

of extensive quantities).

The article [15] by Reyes and the author develop some further differential calculus of ex-

tensive quantities, not only in dimension 1, as here; but it is couched in terms of the Schwartz(double dualization) paradigm, which we presently want to push in the background.

1
http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1http://arxiv.org/abs/1105.3405v1


2/21

1 Monads and their algebras

The relationship between universal algebra, on the one hand, and monads1 on the category

of sets on the other, became apparent in the mid 60s, through the work of Linton, Manes,Kleisli, and many others:

IfT = (T,,) is such a monad, andX is a set, an element P T(X) may be interpretedas an X-ary operation on arbitrary T-algebras B = (B,): if : X B is an X-tuple ofelements in B, we can construct a single element P, B, namely the value of

T(X)T() T(B)

B

on the element P T(X). Then every morphism f : B C of T-algebras is a homomor-phism with respect to the operation defined by P. There is also a converse statement.

The monad-theoretic formulation of universal algebra can be lifted to symmetric

monoidal closed categories E other than sets, provided one considers the monad T to be

E-enriched2, in particular, it applies to E-enriched monads on any cartesian closed cate-

gory E.Recall that for any functor T : E E, one has maps hom(X, Y) hom(T(X), T(Y)),

sending f hom(X, Y) to T(f) hom(T(X), T(Y)); the E-enrichment means that thesemaps not only can be defined for the hom-sets, but for the hom-objects, so that we get maps

(the strength ofT)

stX,Y : YX T(Y)T(X); (1)

YX, as an object ofE carries more structure than the mere setof maps from X to Y, e.g. it

may carry some topology, ifE happens to be of topological nature.

This widening of the scope of monad theoretic universal algebra was documented in a

series of articles by the author in the early 1970s, cf. [6], [7], [8], [9], [10]. In particular, the

formulation makes sense for cartesian closed categories, which is the context of the present

note. Via this formulation, it makes contact with functional analysis, because the basic logic

behind functional analysis is the ability to form function spaces, even non-linear ones. Forinstance, the category E of convenient vector spaces, and the smooth (not necessarily non-

linear maps) is a cartesian closed category, where several theories of functional analysis have

natural formulation, e.g. the theory of distributions (in the sense of Schwartz and others).

We expound here some aspects of the relationship between the theory of strong monads

and functional analysis, by talking about E as if it were just the category of sets, and where

E-enrichment therefore is automatic. (A rigourous account of these aspects for general

cartesian closed categories E is given in [14].) So our exposition technique here is in the

spirit of the naive exposition of synthetic differential geometry, as given in [13], say.

One aim of the theory developed in [14], and expounded synthetically in the present

note, is to document that the space T(X) may be seen as a space ofextensive quantities (ofsome type) on X, in the sense of Lawvere. So we prefer to talk about a P T(X) as anextensive quantity on X, rather than as an X-ary operation, operating on all T-algebras B.

1For the notion of monad, and algebras for a monad, the reader may consult [2].2for these notions, the reader is again referred to [ 2].

2


3/21

To make this work well, one must assume that the monad T is commutative3, see Section 2

and 3.

A main emphasis in Lawveres ideas is that the space of extensive quantities onX depend

(covariant) functorially on X; in the present context, functorality is encoded by the fact thatT is a functor. So for f : X Y, we have T(f) : T(X) T(Y); when T is well understoodfrom the context, we may write f for T(f); this is a type of notation that is as old as thevery notion of functor (recall homology!).

Thus, the semantics of P T(X), as an X-ary operation an T-algebras (B,), may berendered in terms of a pairing

P, := ((P)), (2)

where :XB is a map (anX-tuple of elements ofB). This semantic aspect of extensivequantities is essential in Section 9, where it is seen as the basis of a synthetic theory of

Schwartz distributions.

For completeness, let us indicate how the pairing is defined without using individual

elements P T(X) and BX, as a map

T(X) BX B,

namely utilizing the assumed enrichment (1) ofT over E:

T(X) BXT(X) st

T(X) T(B)T(X)ev T(B)

B;

here, ev denotes the evaluation map (part of the cartesian closed structure ofE). Hence-

forth, we shall be content with using synthetic descriptions, utilizing elements.

We note the following naturality property of the pairing: for f : X Y, P T(X) and : Y B, we have

f(P), = P, f(), (3)

where f() := f. For, the left hand side is (fP), and the right hand side is

((f())P) = (( f)(P)),

but ( f) = f since T is a functor.Similarly, if B = (B,) and (C = (C,) are T-algebras, and F : B C is a T-

homomorphism4, we have, for P T(X) and : X B that

F(P,) = P, F). (4)

This is an immediate consequence of F= T(F), the equation expressing that F is aT-homomorphism.

The terminal object of E is denoted 1. The object T(1) plays a special role as thealgebra of scalars, and we denote it also R; with suitable properties of T, it will in fact

3

It should be stressed that to be commutative is a property of enriched (=strong) monads, and enrichment is astructure, not a property. However, for the case where E is the category of sets, enrichment is automatic.

4later in this article, T-homomorphisms will be called T -linear maps.

3


4/21

carry a canonical commutative ring structure, see Section 7. Its multiplicative unit 1 is the

element picked out by 1. For any X E, we have a unique map X 1, denoted ! (whenX is understood from the context). For P T(X), we have canonically associated a scalar

tot(P) T(1), the total ofP, namely

tot(P) :=!(P).

From uniqueness of maps to 1 follows immediately that P and f(P) have the same total,for any f : X Y.

One example of a monad T (on a suitable cartesian closed category of smooth spaces)

is where T(X) is the space of Schwartz distributions of compact support; we return toSchwartz distributions in 9, and they are only mentioned here as a warning, namely that

functorality is a strong requirement; thus for instance, a uniform distribution on the line can

never have a total. (The functorial properties of non-compact distributions are not under-

stood well enough presently.) In [1], the authors construct the free real vector space monad

T, in a category of suitable convenient spaces ( [16]), by carving it, out by topological

means, from the monad of (compactly supported) Schwartz distributions.The units X : X T(X) will, in terms of Schwartz distribution theory, pick out theDirac distributions x; therefore, we shall allows ourselves the following doubling of nota-tion: for x X, we write

X(x) = x

(with X understood from the context, on the right hand side).

Proposition 1 Let B = (B,) be a T -algebra, and : X B a map. Then

x , = (x).

In particular,

x ,X = x .

Proof. In elementfree terms, the first equation says that the composite

XX T(X)

T() T(B)

B

equals :X B. And this holds, because by naturality of , T()X = B ; but Bis the identity map on B, by the unitary law for the algebra structure . The second equationis then immediate.

An extensive quantity of the form x has total 1 T(1),

tot(x) = 1. (5)

For, the composite map

XX T(X)

T(!) T(1) = R

4


5/21

equals the composite

X! 1

1 T(1) = R, (6)

by naturality of w.r.to ! : X 1, and this is the map taking value 1 for all x X.Proposition 2 For the total of P T(X), we have

tot(P) = P, 1X,

where 1X denotes the function X R with constant value 1 R.

Proof. The map 1X is displayed in (6) above, so P, 1X is by definition the result of applyingto P T(X) the composite

T(X)T(!) T(1)

T(1) T2(1)

1 T(1) = R.

Now by one of the unit laws for a monad, the composite of the two last maps here is the

identity map of T(1), so the displayed composite is just T(!); this is the map which to Preturns the total ofP.

Notation. We attempted to make the notation as standard as possible. On three occasions,

this forces us to have double notation, like the - doubling above, and later E (expecta-tion) for , and an integral symbol for the pairing of extensive and intensive quantities.An exception to standard notation is that the exponential object BX is denoted X B, to

keep it online. (Other online notations have also been used, like [X,B] or XB.)

2 Tensor product of extensive quantities

Let T be an algebraic theory, and let P and Q be an X-ary and a Y-ary operation of it. Then

one can define, semantically, an X Y-ary operation P Q: given an X Y-tuple on theT

-algebraB

; we think of as an matrix of elements ofB

withX

rows andY

columns. Nowevaluate P on each of the Y columns; this gives a Y-tuple of elements ofB; then evaluate Q

on this Y-tuple; this gives an element of B. This element is declared to be the value of the

X Y-ary operation P Q on .One might instead first have evaluated Q on each of the rows, and then evaluated P on

the resulting X-tuple; this would in general give a different result, denoted PQ; the theoryis called commutative if and agree.

This kind of tensor product was formulated monad theoretically, and without reference

to the semantics involving T-algebras, in the authors 1970-1972 papers, so as to be appli-

cable for any strong monad on any cartesian closed category E; it takes the form of two

maps5 natural in X and Y E,

T(X) T(Y)

T(X Y);

5denoted in [6] by X,Y and X,Y, respetively

5


6/21

the monad is called commutative if these two maps agree, for all X and Y. Both and make the functor T a monoidal functor, in particular they satisfy a well known associativity

constraint: (P Q) S = P (Q S), modulo the isomorphisms induced by (XY) Z=

X (Y Z), and similarly for . (The nullary part that goes along with , is a map1 T(1).)Among the equations satsified is (X Y) = XY, which in the notation with

reads: for x X and y Y,x y = (x,y) . (7)

If A = (A,) and C= (C,) are T-algebras, it makes sense to ask whether a map A X C is a T-homomorphism in the first variable, cf. [9]; and similarly it makes sense toask whether a map XA C is a T-homomorphism in the second variable. We shall usethe term T -linearmap as synonymous with T-homomorphism; this allows us to use the

term T-bilinearfor a map A B Cwhich is a T-homomorphism in each of the two inputvariables separately (where A,B, and C are T-algebras), and similarly T-linear in the first

variable, etc.

If f : X Y C is any map into a T-algebra C, it extends uniquely over X Y to amap T(X) Y C which is T-linear in the first variable. Similarly f extends uniquelyover XY to a map X T(Y) C which is T-linear in the second variable. However, amap XY Cdoes not necessarily extend to a T-bilinear T(X) T(Y) C; for this, oneneeds commutativity ofT:

3 Commutative monads

We henceforth consider a commutative monad T = (T,,) on E. The reader may havefor instance the free-abelian-group monad in mind.

From [9], we know that commutativity ofT is equivalent to the assertion that : T(X)T(Y) T(XY) is T-bilinear, for all X and Y. (In the non-commutative case, will onlybe T-linear in the second variable, and will only be T-linear in the first variable.)

Then ifC= (C,) is a T-algebra, any map f :XY Cextends uniquely overX Yto a T-bilinear map T(X) T(Y) C. Since f also extends uniquely over XY to aT-linear T(X Y) C, one may deduce that : T(X) T(Y) T(X Y) is in fact auniversal T-bilinear map out of T(X) T(Y).

If B = (B,) is a T-algebra, and X is an arbitrary object, X B carries a canonicalpointwise T-algebra structure inherited from . In the category of sets, this is just thecoordinatwise T-algebra structure on XB.

Let (A,) and (B,) be T-algebras. IfE has sufficiently many equalizers, there is asubobject A T B of A B, which in the set case consists of those maps A B whichhappen to be T-linear. With T commutative, the subobject A T B A B is in fact asub-T-algebra, cf. [8].

IfA = (A,), B = (B,), and C= (C,) are T-algebras, a map A B Cis T-bilinear

iff its transpose A B

C is T-linear, and factors through the subalgebra B

T C.

6


7/21

In [14], Theorem 1, we prove that for a commutative monad T, the exponential adjoint

of the pairing T(X) (XB) B is a T-bilinear map T(X) (XB) T B; so thereforealso, we have

Theorem 1 The pairing T(X) (XB),

B is T -bilinear, for any T -algebra B.

4 Convolution

If a : X Y Z is any map, and P T(X), Q T(Y), we may form a(P Q) T(Z),called the convolution of P and Q along a, and denoted P a Q. Since is T-bilinear anda = T(a) is T-linear, it follows that P a Q depends in a T-bilinear way on P, Q. In diagram,convolution along a is the composite

T(X) T(Y) T(XY)

T(a) T(Z).

The convolution along the unique map 1 1 1 gives a multiplication on R = T(1), whichis commutative.

A consequence of the naturality of w.r.to the maps ! : X 1 and ! : Y 1 is that

tot(P Q) = tot(P) tot(Q), (8)

where the dot denotes the product in R = T(1). Note that this product is itself (modulo theidentification T(1 1) = T(1)) a tensor product, T(1) T(1) T(1 1) = T(1).

We have, for x X and y Y

x a y = a(x,y). (9)

This follows from (7), together with naturality of w.r.to a, which in -terms reads

T(a)((x,y)) = a(x,y) .

Ifa is an associative operation XX X, it follows from properties of monoidal functorsthat the convolution along a, T(X) T(X) T(X) is likewise associative. If a is commu-tative, commutativity of convolution along a will be a consequence, but here, one uses the

assumption that the monad T is commutative.

5 The space of scalars R := T(1)

The space T(1) plays the role of the ring of scalars, or number line. It has a T-linearstructure, since it is a T-algebra, and it carries a T-bilinear multiplication m, namely

T(1) T(1) T(1 1) = T(1),

7


8/21

which is commutative and associative. The multiplicative unit is picked out by 1 : 1 T(1) = R. - Also R acts on any T(X), by

T(1) T(X)

T(1 X) = T(X), (10)

generalizing the description of multiplication onR; it is denoted just by a dot . This action islikewise T-bilinear, and unitary and associative. These assertions follow from T-bilinearity

of, and the compatibility of with .There areproperties ofT which will imply that T-algebras carry abelian group structure;

in this case, R is a commutative ring, with the above m as multiplication, and any T(X) isan R-module, with T-linear maps being in particular R-linear, see Section 7.

6 Intensive quantities, and their action on extensive quan-

tities

The space R = T(1) is a T-algebra, with a commutative T-bilinear commutative monoidstructure (so in the additive case, Section 7, it is in particular a commutative ring). From

general principles6 follows that for any X, the space XR = X T(1) inherits a T-algebrastructure and a T-bilinear monoid structure; in fact R is a contravariant functor withvalues in the category of monoids whose multiplication is T-bilinear. If f : X Y, the mapfR : YR XR preserves this structure. The map fR is denoted f, and is a kindof companion to the covariant f : T(X) T(Y). In the terminology of Lawvere [18], XRis a space ofintensive quantities on X. Note that T only enters in the form of R = T(1).

The monoid X R = X T(1) acts on any space of the form X T(Y), by a simplepointwise lifting of the action ofT(1) on T(Y), described in (10), (with Y instead ofX):

{X T(1)} {X T(Y)} = X {T(1) T(Y)}X {1,Y}

X T(1 Y)= X T(Y).

(The monoid structure on R = T(1) is a speial case.)

We shall describe an action of the monoid X R on T(X). It has a special case themultiplication of a distribution by a function known from (Schwartz) distribution theory7.

Notationally, we let the action be from the right, and denote it ,

T(X) (XR) T(X).

It is T-linear in the first variable8: it is the 1-T-linear extension over XR of a certainmap X (X R) T(X); in other words, we describe first P for the case where

6it is an aspect of the strength of the monad T that the category ET of T-algebras is E-enriched; it is even

cotensored (cf. [2]) over E; and then XB is the cotensor of the space X with the T-algebra B, for any T-algebra

B. This then in particular applies to B = T(1). For details, see e.g. [14].7also, it is analogous to the cap-product action of the cohomology ring (cup product) on homology, cf. e.g. [4]8it is actually T-bilinear.

8


9/21

P = (x) = x for some x X, and where : X R is any function. Namely, we put

x := (x) x ,

recalling that T(X) carries a (left) action by R = T(1) (of course left and right doesnot make any difference here, since the monoids in question are commutative).

Since T-linearity implies homogeneity w.r.to multiplication by scalars in R, we havein particular that

(x ) = (x) (11)

We shall prove that the action is unitary and associative. The unit of XR is 1X, i.e. thefunction with constant value 1 R. So we should prove P 1X = P. By T-linearity in thefirst variable, it is enough to see it when the input P from T(X) is of the form x, so weshould prove, for the unitary property,

x 1X = x

and similarly, for the associative property, it suffices to prove

(x ) = x ()

for and arbitrary functions X R. The first equation then is a consequence of 1X(x) =1; unravelling similarly the second equation, one sees that the two sides are, respectively

((x) (x)) x (using (11)), and ( )(x), and their equality is a consequence of thepointwise nature (and commutativity) of the multiplication in R.

The following result serves in the Schwartz theory in essence as the definition of the

action of intensive quantities on extensive ones. Recall that the multiplicative monoid ofR acts on any T-algebra of the form T(Y), via 1,Y : T(1) T(Y) T(1 Y) = T(Y). Theaction is temporarily denoted ; for Y = 1, it is just the multiplication on R. This actionextends pointwise to an action ofXR on X T(Y).

Proposition 3 Letbe a function X R, and letbe a function X T(Y). Then for anyP T(X),

P , = P, T(Y).

Proof. Since both sides of the claimed equation depend in a T-linear way on P, it suffices

to prove the equation for the case where P is x for some x X. We calculate the left handside:

x , = (x) x , = (x) x , = (x) (x),

using Proposition 1, and the right hand side similarly calculates

x , = ( )(x),

which is likewise (x) (x), because of the pointwise character of the action on XT(Y).

9


10/21

Corollary 1 The pairing P, (for P T(X) and XR) can be described in terms of as follows:

P, = tot(P ).

Proof. Take Y = 1 (so T(Y) = R), and take = 1X. Then

tot(P ) = P , 1X = P, 1X

using (2), and then the Proposition. But 1X = .

7 Additive structure

We shall in the present Section describe a simple categorical property of the monad T, which

will guarantee that T-linearity implies additivity, even R-linearity in the sense of a rig

R E (rig= commutative semiring), namely R = T(1). This condition will in fact implythat ET is an additive (or linear) category.

We begin with some standard general category theory, namely a monad T = (T,,) ona category which has finite products and finite coproducts. (No distributivity is assumed.)

So E has an initial object /0. IfT(/0) E is a terminal object, then the object (T(/0),/0) isa zero object in ET, i.e. it is both initial and terminal. It is initial because T, as a functor

E ET, is a left adjoint, hence preserves initials; and since T(/0) = 1, it is also terminal (theterminal object in ET being 1 E, equipped with the unique map T(1) 1 as structure).This zero object in ET we denote 0. Existence of a zero object in a category implies that

the category has distinguished zero maps 0A,B : A B between any two objects A and B,namely the unique map A B which factors through 0. For ET, we can even talk aboutthe zero map 0X,B : X B, where X E and B = (B,) ET, namely 0X,B is X followedby the zero map 0T(X),B : T(X) B. We have a canonical map X+Y T(X) T(Y): thecomposite X X+Y T(X) T(Y) is (X, 0X,T(Y)) (here, the first map is the coproductinclusion map ). Similarly, we have a canonical map Y T(X) T(Y). Using the universal

property of coproducts, we thus get a canonical mapX,Y :X+Y T(X) T(Y). It extendsuniquely over X+Y : X+Y T(X+Y) to a T-linear map

X,Y : T(X+ Y) T(X) T(Y),

and is natural in X and in Y. We say that T : E ET takes binary coproducts to productsifX,Y is an isomorphism (in E or equivalently in E

T) for all X, Y in E . Note that the

definition presupposed that T(/0) = 1; it is the zero object in ET, so that if T takes binarycoproducts to products, it in fact takes finite coproducts to products, in a similar sense. So

we can also use the phrase T takes finite coproducts to products for this property ofT.

We define an addition map in ET ; it is a map + : T(X) T(X) to T(X), namely thecomposite

T(X) T(X)

1

X,Y

T(X+X)

T()

T(X)

10


11/21

where : X+X X is the codiagonal. So in particular, if ini denotes the ith inclusion(i = 1, 2) ofX into X+X, we have

idT X =

T XT(ini)

T(X+X)X,X

T X T X+

T X

. (12)

Note that this addition map is T-linear. Under the identification T(X) = T(X+ /0) = T(X)1, the equation (12) can also be read: T(!) : T(/0) T(X) is right unit for +, and similarlyone gets that it is a left unit.

We leave to the reader the easy proof of associativity and commutativity of the map

+ : T(X) T(X) T(X). It follows that T(X) acquires structure of an abelian monoid inET (and also in E).

For an abelian monoid A in any category, we may ask whether A is an abelian group or

not (so there is a minus corresponding to the +); existence of such minus is a propertyof A, not an added structure. If T is a monad which takes finite coproducts to products, it

makes sense to ask whether the canonical monoid structure which T-algebras in this case

have, is actually an abelian group structure; it is therefore a property on such T, not an added

structure. We shall henceforth assume this property, since we need minus (= difference)for differential calculus.

In [14], we proved that

Proposition 4 Every T-linear map T(X) T(Y) is compatible with the abelian groupstructure.

We again assume that T is a commutative monad. Recall that we then have the T-

bilinear action T(X) T(1) T(X). It follows from the Proposition that it is additive ineach variable separately.

We have in particular the T-bilinear commutative multiplication m : T(1) T(1) T(1), likewise bi-additive, m(x +y,z) = m(x,z) + m(y,z), or in the notation one also wantsto use,

(x +y) z = x z +y z,

so that R = T(1) carries structure of a commutative ring. We may summarize:

Proposition 5 Each T(X) is a module over the ring R = T(1); each T-linear map T(X) T(Y) is an R-module morphism.

It is more generally true that T-linear maps A B (for A and B ET) are R-module maps.We shall not use this fact.

The property of T that it takes finite coproducts to products accounts for a limited

aspect of contravariance for extensive quantities: if a space X is a coproduct, X = X1 +X2,the isomorphism T(X1 +X2) = T(X1) T(X2) implies that an extensive quantity P T(X)gives rise to a pair of extensive quantities P1 T(X1) and P2 T(X2), which one may

reasonably may call therestrictions

ofP

toX

1 andX

2, respectively. Now, restriction is acontravariant construction, and applies as such, for intensive quantities, along any map. For

11


12/21

extensive quantities, and for monads T of the special kind studied here, it applies only to

quite special maps, namely to the inclusion maps into finite coproducts, like X1 X1 +X2.We leave to the reader to philosophize over the extent to which, given a distribution of

smoke in a given room, it makes sense to talk about the distribution of this quantity ofsmoke, restricted to the lower half of the room.

(For distributions in the Schwartz sense, one may construct some further restriction

constructions (restriction to open subsets); this is an aspect of the fact that the corresponding

intensive quantities (= smooth functions) admit an extension construction from closed

subsets.)

For u R, we have the translation map u : R R given by x x + u. IfP T(R), wehave thus also u (P) T(R).

We have the following reformulation of the translation maps in terms of convolution

along the addition map + : R R R:

Proposition 6 For any P T(R) and a R,

a

(P) = a P = P a.

Proof. The second equation follows from commutativity of+. To see the first equation, weobserve that both sides ofa (P) = a P depend T-linearly on P; so it suffices to prove thisequation for the case where P is of the form b for b R. But

a (b) = a+b = b+a.

In particular, we see that 0 is a neutral element for convolution, 0 P = P 0 = P.

8 Differential calculus of extensive quantities on R

We attempt in this Section to show how some differential calculus of extensive quantities T(R) may be developed on equal footing with the standard differential calculus of intensivequantities (meaning here: functions defined on R). For this, we assume that the monad T

on E has the properties described in Section 7, so in particular, R is a commutative ring. To

have some differential calculus going for such R, one needs some further assumption.Consider a commutative ring R. Assume D R is a subset satisfying the following

KL-axiom:

for any f : R R, there exists a unique f : R R such that for all x R

f(x + d) = f(x) + d f(x) for all d D. (13)

Example: 1) models of synthetic differential geometry, with D the set of d R withd2 = 0 (the simple Kock-Lawvere axiom says (cf. e.g. [11]) a little more than this, namelyit also asks that any function f : D R extends to a function f : R R.)

2) Any commutative ring, with D = {d} for one single invertible d R. In this case, forgiven f, the f asserted by the axiom is the function

f(x) =1

d (f(x + d) f(x)),

12


13/21

the standard difference quotient.

Similarly, ifV is an R-module, we say that it satisfies KL, if for any f : R V, thereexists a unique f : R V such that (13) holds for all x R.

In either case, we may call f

the derivative of f.It is easy to see that any commutative ringR is a model, using {d} as D, as in Example 2)

(and then also, anyR-module V satisfies then the axiom); this leads to some calculus of finite

differences. Also, it is true that ifE is the category of abstract sets, there are no non-trivial

models of the type in Example 1); but, on the other hand, there are other cartesian closed

categories E (e.g. certain toposes containing the category of smooth manifolds, cf. e.g. [11]),

and where a rather full fledged differential calculus for intensive quantities emerges from

the KL-axiom.

We assume that R = T(1) satisfies the KL-axiom (for some fixed D R), and also thatany R-module of the form T(X) does so.

Proposition 7 (Cancelling universally quantified ds) If V is an R-module which satisfies

KL, and v V has the property that d v = 0 for all d D, then v = 0.

Proof. Consider the function f : R V given by t t v. Then for all x R and d D

f(x + d) = (x + d) v = x v + d v,

so that the constant function with value v will serve as f. On the other hand, d v = d 0 byassumption, so that the equation may be continued,

= x v + d 0

so that the constant function with value 0 V will likewise serve as f. From the uniquenessof f, as requested by the axiom, then follows that v = 0.

We are now going to provide a notion ofderivative P for any P T(R). Unlike differen-tiation of distributions in the sense of Schwartz, which is defined in terms of differentiation

of test functions , our construction does not mention test functions, and the Schwartz def-inition P , := P, comes in our treatment out as a result, see Proposition 13 below.

For u = 0, Pu (P) = 0 T(R). Assuming that theR-module T(R) is KL, we thereforehave for any P T(R) that there exists a unique P T(R) such that for all d D,

d P = P d (P).

Since d P has total 0 for all d D, it follows that P has total 0.Differentiation is translation-invariant: using

t s = t+s = s t,

it is easy to deduce that

(t(P)) = (

t

)(P). (14)

13


14/21

Proposition 8 Differentiation of extensive quantities on R is a T -linear process.

Proof. Let temporarily : T(R) T(R) denote the differentiation process. Consider afixed d D. Then for any P T(R), d(P) = d P is P d

P; it is a difference of the two

T-linear maps, namely the identity map on T(R) and d = T(d), and as such is T-linear.

Thus for each d D, the map d : T(R) T(R) is T-linear. Now to prove T-linearity of means, by monad theory, to prove equality of two maps T2(R) T(R); and since dis T-linear,as we proved, it follows that the two desired maps T2(R) T(R) become equalwhen post-composed with the map multiplication by d: T(R) T(R). Since d D wasarbitrary, it follows from KL axiom for the R-module T(X) that the two desired maps areequal, proving T-linearity.

The structure map T(R) R of the T-algebra R = T(1) is 1 : T2(1) T(1). Just

as plays a special role, with (x) being the Dirac distribution x, the structure mapsfor T-algebras play a role that sometimes deserves an alternative notation and name; thus

in particular 1 : T(R) R plays in the context of probability distributions the role ofexpectation, see [14], and we shall here again allow ourselves a doubling of notation and

terminology:E(P) := 1(P),

the expectation ofP T(R). It is a scalar R.Note that for a R,

E(a) = a; (15)

since a is R(a) = T(1)(a), and E = 1, this is a consequence of the monad law thatX T(X) is the identity map ofT(X) for any X, in particular for X = 1.

Proposition 9 Let P T(R). Then

E(P) = tot(P).

Proof. The Proposition say that two maps T(R) R agree, namely E and tot, where, as above, is the differentiation process P P. Both these maps are T-linear, so it sufficesto prove that the equation holds for the case P = x, so we should prove

E(x) = tot(x).

By the principle of cancelling universally quantified ds (Proposition 7), it suffices to prove

that for all d D thatdE(x) = d tot(x).

The right hand side is d, by (5). The left hand side is

E(dx) = E(x d x)

= E(x x+d)= E(x) E(x+d) = x (x + d) = d,

14


15/21

by (15). This proves the Proposition.

The differentiation process for functions, as a map R V R V, is likewise T-linear,but this important information cannot be used in the same way as we used T-linearity of the

differentiation T(R) T(R), since, unlike T(R), R V (not even R R) is not known to befreely generated by elementary quantities like the xs.

Recall that ifF : V W is an R-linear map between KL modules

F = (F) (16)

for any : R V.

One can generalize the differentiation of extensive quantities on R to a differentiation of

extensive quantities on any space X equipped with a vector field. The case made explicit is

where the vector field is (x, d) x + d (or x ) on R.

Proposition 10 Let P T(R) and Q T(R). Then

(P Q) = P Q = P Q.

Proof. By commutativity of convolution, it suffices to prove that (P Q) = P Q. Bothsides depend in a T-bilinear way on P and Q, so it suffices to see the validity for the case

where P = a and Q = b. To prove (a b) = a b, it suffices to prove that for all dD,

d (a b) = da b,

and both sides comes out as a+b a+b+d, using that is R-bilinear.

Primitives of extensive quantities

We noted already in Section 1 that P and f(P) have same total, for any P T(X) and

f :X Y. In particular, for P T(R) and dD, dP = Pd (P) has total 0, so cancellingthe universally quantified d we get that P has total 0.

A primitive of an extensive quantity Q T(R) is a P T(R) with P = Q. Since any P

has total 0, a necessary condition that an extensive quantity Q T(R) has a primitive is thattot(Q) = 0. Recall that primitives, in ordinary 1-variable calculus, are also called indefiniteintegrals, whence the following use of the word integration:

Integration Axiom. Every Q T(R) with tot(Q) = 0 has a unique primitive.

(For contrast: for intensive quantities on R (so : R R is a function), the standardintegration axiom is that primitives always exist, but are notunique, only up to an additive

constant.)

By R-linearity of the differentiation process T(R) T(R), the uniqueness assertion inthe Axiom is equivalent to the assertion: if P = 0, then P = 0. (Note that P = 0 implies that

P is invariant under translations d (P) = P for all d D.) The reasonableness of this latterassertion is a two-stage argument: 1) if P = 0, P is invariant under arbitary translations

15


16/21

u (P) = P. 2) ifP is invariant under all translations, and has compact support, it must be 0.(Implicitly here is: R itself is not compact.)

In standard distribution theory, the Dirac distribution a (where a R) has a primitive,

namely the Heaviside function; but this function has not compact support - its supportis a half line R.

On the other hand, the integration axiom provides a (unique) primitive for a distribution

of the form a b, with a and b in R. This primitive is denoted [a, b], the interval from ato b; thus, the defining equation for this interval is

[a, b] = a b.

Note that the phrase interval from . . . to . . . does not imply that we are considering an

ordering on R (although ultimately, one wants to do so).

Proposition 11 The total of[a, b] is b a.

Proof. We have

tot([a, b]) = E([a, b]) = E(a b)

by Proposition 9 and the fact that [a, b] is a primitive ofa b

= E(a) +E(b) = b a,

by (15).

It is of some interest to study the sequence of extensive quantities

[a, a], [a, a] [a, a], [a, a] [a, a] [a, a], . . . ;

they have totals 2a, (2a)2, (2a)3, . . .; in particular, if 2a = 1, this is a sequence of probabilitydistributions, approaching a Gauss normal distribution (the latter, however, has presently no

place in our context, since it does not have compact support).

9 Extensive quantities and Schwartz distributions

Recall from (2) that P T(X) gives rise to an X-ary operation on any T-algebra B = (B,),via P, := ((P)), where XB. The pairing is thus a map

T(X) (XB),

B.

which is T-bilinear (cf. Theorem 1, or [14]). We may take the exponential transpose of the

pairing; this is then a map X : T(X) (XB) T B.The synthetic rendering of Schwartz distribution theory is that (X R) T R is the

space of Schwartz distributions of compact support: X R is the space of test functions

(not necessarily of compact support), and (XR) T R is the space ofT-linear functionals

16


17/21

X R R on the space of such test functions. (In some well adapted models E of SDG,and for suitable T, this can be proved to be an object whose set of global sections is in fact

the standard Schwartz distributions of compact support on X, if X is a smooth manifold; cf.

[19] Proposition II.3.6 (Theorem of Que and Reyes).)Remark. Consider a commutative ring object in a sufficiently cocomplete cartesian closed

category E. Let T be the (strong) monad which to X associates the free R-module on X.

Thus in particular R = T(1). The monadX (XR)T R is in general not a commutativemonad, so cannot agree with T, although in some cases, the monad map : T (R)TR is monic. In [1], it is proved that T(X) for a special case (convenient vector spaces) canbe carved out of (X R) T R by topological means. Other investigations, e.g. in [12],and a Theorem of Waelbroeck, describe a class of spaces X which perceive X to be anisomorphism.

To make contact with classical theory and intuition, we introduce, for the third time,

a doubling of notation (this one is actually quite classical); for P T(X) and X B(where B is a T algebra), we write

X(x) dP(x) := P, B,

with x a dummy variable ranging over X. Thus Proposition 2 may be rendered

tot(P) =

X

1 dP(x).

For B = T(1) =R and P T(R) = T2(1), the

-notation will help to motivate the use of the

terminology expectation ofP for 1(P); let : R R be the identity map. Then

P, = 1((P)) = 1(P),

since

is the identity map of T(R

). On the other hand

X(x) dP(x) =

X

x dP(x),

since (x) = x; this is the standard integral expression for expectation E(P) for a proba-bility distribution P on R.

The map X : T(X) (X R) T R is not necessarily monic; in the case of classicalSchwartz distributions, it is monic, which allows the classical theory to identify extensive

quanties in T(X) with elements in (X R) T R, and so one avoids having to mentionT(X) explicitly; the notion of extensive quantity on X is thus made dependent on the notionof intensive quantity (test function) on X. It is, however, easy to give examples of Ts

where there are not sufficiently many test functions X R (with R = T(1)) to make

X : T(X) (X

R)

T R injective, whence one motivation for the study ofT, independentof the introduction ofXR.

17


18/21

The injectivity ofX may be expressed: To test equality ofP and Q in T(X), it sufficesto test, for arbitrary functions : X R, whether P, = Q,, whence the name testfunction.

If X is not injective, there are not sufficiently many such test functions X R, butthere are enough, if we allow test functions with arbitrary T-algebras B = (B,) as theircodomain. We have in fact

Proposition 12 For any X E, there exists a T-algebra B so thatX : X (XB) T Bis monic.

Proof. Take B = T(X); then the map e : (X T(X))T T(X) T(X) given by evaluationat X : X T(X) is left inverse for X. For, ifP T(X), then to say e(X(P)) = P isequivalent to saying

P,X = P. (17)

Since the structure map of the T-algebra T(X) is X, the definition (2) of the pairing P,Xgives P,X = X((P)) (where here is short for X). However, is just anothernotation for T

(), and

X T

(X)is the identity map on T

(X

)by one of the monad laws.

So we get P back when we apply this map to P.

Thus, instead of identifying an extensive quantity on X by its action on arbitrary test

functions : X R, we identify it by its action on one single test function, namely thefunction X : X T(X).

I conjecture that T(X) is actually the end

BET(XB) T B.

Here is an important relationship between differentiation of extensive quantities on R,

and of functions : R T(X); such functions can be differentiated, since T(X) is assumedto be KL as an R-module. (In the Schwartz theory, this relationship, with X = 1, serves asdefinition of derivative of distributions.)

Proposition 13 For P T(R) and R T(X), one has

P, = P,.

Proof. We are comparing two maps T(R) (R T(X)) T(X), both of which are T-linearin the first variable. Therefore, it suffices to prove the equality for the case ofP = t; in fact,by R-bilinearity of the pairing, it suffices to prove that for any t R and d D, we have

d (t)

, = t, d.

The left hand side is t d (t),, and using bi-additivity of the pairing, this gives(t)((d))()(t) = (t) (t+ d), which is d(t).

Proposition 9 can be seen as a special case, with X = 1 (thus T(X) = R), and with the identity function R R. We use the integral notation. Thus x denotes the identityfunction on R. So

E(P) =

Rx dP(x) =

R

(x) , dP(x) =

R1 dP(x),

18


19/21

the middle equality by Proposition 13. This, however, is tot(P), by Proposition 2.

The relationship between differentiation of extensive and intensive quantities on R ex-

pressed in Proposition 13 may be given a more compact formulation, using (17). For then

we have, for P T(R), thatP = P ,R = P,

R.

Thus in particular, knowledge of R gives knowledge of P for any P T(R). It also gives

knowledge of for any : R V, with V a T-algebra which is KL module. For, any such extends over R to a (unique) T-linear F : T(R) V, so = FR; therefore

= (FR) = F R,

using (16).

Let us calculate R : R T(R) explicitly; we have for any d D and x R (writing for R)

d (x) = (x + d) (x) = x+d x = d (x)

,

so cancelling the universally quantified d, we get for any x R that (x) = (x)

.

The first differentiation refers to differentiation of functions, the second to differentiation

of distributions; it is tempting to write the former with a Newton dot; then we get (x) =

(x).

The following depends on the Leibniz rule for differentaiating a product of two func-

tions; so this is notvaliud under he general assumptions of this Section, but needs the further

assumption of Example 2, namely thet D consists ofd R with d2 = 0, as in synthetic dif-ferential geometry. We shall then use test function technique to prove

Proposition 14 For any P T(R) and R R,

(P )

= P

+ P

.

Proof. It suffices to prove that for the universal test function = X : X T(X), wehave

(P ) , = P , + P ,.

We calculate:

(P ) , = P , (by Proposition 13)

= P, (by Proposition 3)

= P, ( )

using that Leibniz rule applies to any bilinear pairing, like ,

= P,

( )

+ P,

= P , + P,

19


20/21

using Proposition 13 on the first summand

= P , + P ,

using Proposition 3 on each summand

= P + P ,

In other words, (replacing by ), the proof looks formally like the one from books ondistribution theory, but does not depend on sufficiently many test functions with values in

R.

References

[1] R. Blute, T. Ehrhard and C. Tasson, A convenient differential category, to appear in

Cahiers de Top. et Geom. Diff. Cat.

[2] F. Borceux, Handbook of Categorical Algebra Vol. 2, Cambridge university Press

1994.

[3] S. Eilenberg and M. Kelly, Closed Categories, Proc. Conf. Categorical Algebra La

Jolla 1965, 421-562, Springer Verlag 1966.

[4] P.J. Hilton and S. Wylie, Homology Theory, Cambridge University Press 1960.

[5] M. Kelly, Basic Concepts of Enriched Category Theory, London Math. Soc. Lecture

Notes 64, Cambridge University Press 1982.

[6] A. Kock, Monads on symmetric monoidal closed categories. Arch. Math. (Basel),

21:110, 1970.

[7] A. Kock, On double dualization monads. Math. Scand., 27:151165, 1970.

[8] A. Kock, Closed categories generated by commutative monads. J. Austral. Math. Soc.,

12:405424, 1971.

[9] A. Kock, Bilinearity and Cartesian closed monads. Math. Scand., 29:161174, 1971.

[10] A. Kock, Strong functors and monoidal monads. Arch. Math. (Basel), 23:113120,

1972.

[11] A. Kock, Synthetic Differential Geometry, London Math. Soc. Lecture Notes 51, Cam-

bridge University Press 1981; Second Edition London Math. Soc. Lecture Notes 333,

Cambridge University Press 2006.

[12] A. Kock, Some problems and results in synthetic functional analysis, Category Theo-

retic Methods in Geometry Aarhus 1983, Aarhus Var. Publ. Series 35 168-191, 1983.

20


21/21

[13] A. Kock, Synthetic Geometry of Manifolds, Cambridge Tracts in Math. 180, Cam-

bridge University Press 2010.

[14] A. Kock, Monads and extensive quantities, arXiv [math.CT] 1103.6009

[15] A. Kock and G.E. Reyes, Some calculus with extensive quantities, TAC 11 (2003),

321-336.

[16] A. Kriegl and P. Michor, The Convenient Setting of Global Analysis, Am. Math. Soc.

1997.

[17] F.W. Lawvere, Algebraic Concepts on the Foundations of Physics and Engineering,

MTH 461/561, Buffalo Jan. 1987.

[18] F.W. Lawvere, Categories of space and of quantity, in: J. Echeverria et al. (eds.), The

Space of Mathematics , de Gruyter, Berlin, New York (1992)

[19] I. Moerdijk and G.E. Reyes, Models for Smooth Infinitesimal Analysis, Springer 1991.

[20] L. Schwartz, Methodes mathematiques pour les sciences physiques, Hermann Paris

1961.

[email protected]

21

calculus of extensive quantities

Documents